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| Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| distgp.1 | ⊢ 𝐵 = (Base‘𝐺) | 
| distgp.2 | ⊢ 𝐽 = (TopOpen‘𝐺) | 
| Ref | Expression | 
|---|---|
| distgp | ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp) | |
| 2 | simpr 484 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵) | |
| 3 | distgp.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V | 
| 5 | distopon 23004 | . . . . 5 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵)) | |
| 6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ 𝒫 𝐵 ∈ (TopOn‘𝐵) | 
| 7 | 2, 6 | eqeltrdi 2849 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵)) | 
| 8 | distgp.2 | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐺) | |
| 9 | 3, 8 | istps 22940 | . . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵)) | 
| 10 | 7, 9 | sylibr 234 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp) | 
| 11 | eqid 2737 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
| 12 | 3, 11 | grpsubf 19037 | . . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) | 
| 13 | 12 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) | 
| 14 | 4, 4 | xpex 7773 | . . . . 5 ⊢ (𝐵 × 𝐵) ∈ V | 
| 15 | 4, 14 | elmap 8911 | . . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵) | 
| 16 | 13, 15 | sylibr 234 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵))) | 
| 17 | 2, 2 | oveq12d 7449 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵)) | 
| 18 | txdis 23640 | . . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)) | |
| 19 | 4, 4, 18 | mp2an 692 | . . . . . 6 ⊢ (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵) | 
| 20 | 17, 19 | eqtrdi 2793 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵)) | 
| 21 | 20 | oveq1d 7446 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽)) | 
| 22 | cndis 23299 | . . . . 5 ⊢ (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) | |
| 23 | 14, 7, 22 | sylancr 587 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) | 
| 24 | 21, 23 | eqtrd 2777 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵))) | 
| 25 | 16, 24 | eleqtrrd 2844 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | 
| 26 | 8, 11 | istgp2 24099 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) | 
| 27 | 1, 10, 25, 26 | syl3anbrc 1344 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Basecbs 17247 TopOpenctopn 17466 Grpcgrp 18951 -gcsg 18953 TopOnctopon 22916 TopSpctps 22938 Cn ccn 23232 ×t ctx 23568 TopGrpctgp 24079 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-0g 17486 df-topgen 17488 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-tmd 24080 df-tgp 24081 | 
| This theorem is referenced by: (None) | 
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